Relations between kinetic parameters of different column models for liquid chromatography applying core-shell particles

Figures & data

Table 1. Analytical moments of GRM, LKM and EDM for core-shell particles (except $\mu {\prime }_4$ of GRM). Here, the zeroth moment ${\mu }_0=c_{ \text{inj}}{\tau }_{ \text{inj}}$, representing the amount injected, is the same for all models.

Moments	GRM	LKM (or EDM when $k_{\text{LKM}}\to \infty$ and $\mathit{Pe}=\frac{\mathit{Lu}}{D_{\text{app}}}$)
μ1	$$\frac{{\tau }_{\text{inj}}}{2}+\left(1+{\tilde{a}}^{*}{F}_e\right)$$	$$\frac{{\tau }_{\text{inj}}}{2}+\left(1+\tilde{a}{F}_e\right)$$
$\mu {\prime }_2$	$$\frac{{\tau }_{\text{inj}}^2}{12}+\frac{2}{\mathit{Pe}}{\left(1+{\tilde{a}}^{*}{F}_e\right)}^2+2F_e{({\tilde{a}}^{*})}^2\left[\frac{F_e}{\xi }+\frac{1}{15{\eta }_{\text{mod}}}\right]$$	$$\frac{{\tau }_{\text{inj}}^2}{12}+\frac{2}{\mathit{Pe}}{\left(1+\tilde{a}{F}_e\right)}^2+\frac{2\tilde{a}{F}_e^2{ϵ}_e}{{\kappa }_{\text{LKM}}}$$
$\mu {\prime }_3$	$$\frac{12}{P{e}^2}{\left(1+{\tilde{a}}^{*}{F}_e\right)}^3+\frac{6(1+{\tilde{a}}^{*}{F}_e)F_e}{\mathit{Pe}}\left[\frac{2F_e{({\tilde{a}}^{*})}^2}{\xi }+\frac{2{({\tilde{a}}^{*})}^2}{15{\eta }_{\text{mod}}}\right]$$	$$\frac{12}{P{e}^2}{\left(1+\tilde{a}{F}_e\right)}^3+\frac{12}{\mathit{Pe}}\frac{\left(1+\tilde{a}{F}_e\right){ϵ}_e\tilde{a}{F}_e^2}{{\kappa }_{\text{LKM}}}+\frac{6{ϵ}_e^2\tilde{a}{F}_e^3}{{\kappa }_{\text{LKM}}^2}$$
	$$+F_e{({\tilde{a}}^{*})}^3\left[\frac{4}{105{\eta }_{\text{mod}}^2}\frac{(1+{\rho }_2)}{{(1+{\rho }_1)}^2}+\frac{4F_e}{5\xi {\eta }_{\text{mod}}}+\frac{6F_e^2}{{\xi }^2}\right]$$
$\mu {\prime }_4$	$$\frac{12{\left(1+a^{*}{F}_e\right)}^4}{P{e}^4}\left[P{e}^2+10Pe\right]-\frac{4}{175}\frac{F_e^2{a}^{*4}\left(2+\mathit{Pe}\right)}{{\eta }^{2}Pe}$$	$$\frac{{\tau }_{\text{inj}}^4}{80}+{\tau }_{\text{inj}}^2\left[\frac{1}{\mathit{Pe}}{(1+\tilde{a}{F}_e)}^2+\frac{{ϵ}_e\tilde{a}{F}_e^2}{{\kappa }_{\text{LKM}}}\right]$$
	$$+\frac{24{(1+a^{*}{F}_e)}^2{F}_e{a}^{*2}}{P{e}^3}\left(\frac{F_e}{\xi }+\frac{1}{15\eta }\right)\left[P{e}^2+6Pe\right]$$	$$+\frac{12{\left(1+\tilde{a}{F}_e\right)}^4}{P{e}^4}\left[P{e}^2+10Pe\right]+\frac{24{ϵ}_e\tilde{a}{F}_e^2{\left(1+\tilde{a}{F}_e\right)}^2}{P{e}^3{\kappa }_{\text{LKM}}}\left[P{e}^2+6Pe\right]$$
	$$+\frac{2F_e{a}^{*3}}{\mathit{Pe}}\left(\frac{4}{105{\eta }^2}+\frac{4F_e}{5\xi \eta }+\frac{6F_e^2}{{\xi }^2}\right)\left[4+a^{*}{F}_e(6+\mathit{Pe})\right]$$	$$+\frac{12{ϵ}_e{}^2\tilde{a}{F}_e^3}{\mathit{Pe}{\kappa }_{\text{LKM}}^2}\left[4+\tilde{a}F(6+\mathit{Pe})\right]+\frac{24{ϵ}_e{}^3\tilde{a}{F}_e^4}{{\kappa }_{\text{LKM}}^3}$$
	$$+8F_e{a}^{*4}\left(\frac{1}{525}\frac{1}{{\eta }^3}+\frac{9}{175}\frac{F_e}{{\eta }^2\xi }+\frac{3}{5}\frac{F_e^2}{\eta {\xi }^2}+\frac{3F_e^3}{{\xi }^3}\right)$$
	$$+\frac{{\tau }_{\text{inj}}^4}{80}+{\tau }_{\text{inj}}^2\left[\frac{{(1+a^{*}{F}_e)}^2}{\mathit{Pe}}+F_e{a}^{*2}\left(\frac{F_e}{\xi }+\frac{1}{15\eta }\right)\right]$$
	(valid only for fully-porous particles)

$${\eta }_{\text{mod}}=\frac{\eta }{{\rho }_{\text{mod}}},\hspace{1em}{\rho }_{\text{mod}}=\frac{1+2{\rho }_{\text{core}}+3{\rho }_{\text{core}}^2-{\rho }_{\text{core}}^3-5{\rho }_{\text{core}}^4}{{(1+{\rho }_{\text{core}}+{\rho }_{\text{core}}^2)}^2},\hspace{1em}{\rho }_1=\frac{{\rho }_{\text{core}}(1+3{\rho }_{\text{core}}+3{\rho }_{\text{core}}^2-{\rho }_{\text{core}}^3)}{{(1+{\rho }_{\text{core}}+{\rho }_{\text{core}}^2)}^2},\hspace{1em}{\rho }_2=\frac{{\rho }_{\text{core}}(2+9{\rho }_{\text{core}}+\frac{35}{2}{\rho }_{\text{core}}^2+\frac{23}{2}{\rho }_{\text{core}}^3-3{\rho }_{\text{core}}^4-{\rho }_{\text{core}}^5)}{{(1+{\rho }_{\text{core}}+{\rho }_{\text{core}}^2)}^3},$$

$${\tilde{a}}^{*}=[{ϵ}_p+(1-{ϵ}_p\left)a\right](1-{\rho }_{\text{core}}^3),\hspace{1em}\tilde{a}=a(1-{\rho }_{\text{core}}^3),F_e=\frac{1-{ϵ}_e}{{ϵ}_e},\mathit{Pe}=\frac{\mathit{Lu}}{D_z},\mathit{Bi}=\frac{k_{\text{ext}}{R}_p}{D_{\text{eff}}},\eta =\frac{D_{\text{eff}}L}{R_p^{2}u},\xi =3F_e\frac{k_{\text{ext}}}{R_p}\frac{L}{u}.$$

Figure 1. Schematic diagrams of a fixed-bed column packed with inert-core adsorbents.

Table 2. Parameters assumed in simulations.

Parameters	values
Column length	$L=10\text{cm}$
External porosity	${ϵ}_e=0.4$
Internal porosity	${ϵ}_p=0.333$
Henry’s constant	$a=2.5$
Initial concentration	$c_{\text{init}}=0$
Injected bulk concentration	$c_{\text{inj}}=0.5\mathrm{g}/\mathrm{l}$
Injection time	$t_{\text{inj}}=10$
Core-radius fraction	${\rho }_{\text{core}}=0.5$
Axial dispersion coefficient(GRM, LKM)^a	$D_z=0.02{\text{cm}}^2/\text{min}$
Effective diffusivity coefficient (GRM)	$D_{\text{eff}}={10}^{-6}{\text{cm}}^2/\text{min}$
External mass transfer coefficient (GRM)^a	$k_{\text{ext}}=0.01\text{cm}/\text{min}$
Mass transfer coefficient of LKM^a	$k_{\text{LKM}}=1{\text{min}}^{-1}$
Apparent dispersion coefficient of EDM^a	$D_{\text{app}}=0.02{\text{cm}}^2/\text{min}$

^aAssumed to be the same for different flow rates.

Figure 2. Illustration of three models utilizing the non-matched kinetic parameters of Table 2. The effect of ${\rho }_{ \text{core}}$ is shown on the concentration profiles for $u=1 \text{cm}/\text{min}$.

Figure 3. Moments and concentration profiles of the considered three models for ${\rho }_{\text{core}}=0.5$ utilizing the non-matched kinetic parameters of Table 2.

Figure 4. Matching of the second central moments of simpler models with GRM (c.f. Eqs. (17)–(19)) for ${\rho }_{\text{core}}=0.5$. The kinetic parameters of GRM are taken from Table 2.

Figure 5. Matching of the third central moments of simpler models with GRM (c.f. Eqs. (20)–(24)) for ${\rho }_{\text{core}}=0.5$. The kinetic parameters of GRM are taken from Table 2.

Figure 6. Simultaneously matching the first three moments of simpler models with GRM (c.f. Eqs. (26)–(28)) for ${\rho }_{\text{core}}=0.5$. The kinetic parameters of GRM are taken from Table 2.

Figure 7. Matching of GRM moments with the known moments of LKM (${\rho }_{\text{core}}=0$). The kinetic parameters of LKM are taken from Table 2.

Figure 8. Effect of ${\rho }_{\text{core}}$ on moments matching (illustrated for $u=1 \text{cm}/\text{min}$). Parameters are selected to match moments for ${\rho }_{\text{core}}=0.5$. The kinetic parameters of LKM are taken from Table 2.